Integrand size = 42, antiderivative size = 20 \[ \int \frac {1875-3 e+e^{2 x} (1252-2 e-4 x)}{390625+e^2-2500 x+4 x^2+e (-1250+4 x)} \, dx=\frac {-e^{2 x}-3 x}{-625+e+2 x} \]
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Time = 0.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6873, 27, 6874, 2228} \[ \int \frac {1875-3 e+e^{2 x} (1252-2 e-4 x)}{390625+e^2-2500 x+4 x^2+e (-1250+4 x)} \, dx=\frac {e^{2 x}}{-2 x-e+625}+\frac {3 (625-e)}{2 (-2 x-e+625)} \]
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Rule 27
Rule 2228
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {1875 \left (1-\frac {e}{625}\right )+e^{2 x} (1252-2 e-4 x)}{(-625+e)^2-4 (625-e) x+4 x^2} \, dx \\ & = \int \frac {1875 \left (1-\frac {e}{625}\right )+e^{2 x} (1252-2 e-4 x)}{(-625+e+2 x)^2} \, dx \\ & = \int \left (-\frac {3 (-625+e)}{(-625+e+2 x)^2}-\frac {2 e^{2 x} (-626+e+2 x)}{(-625+e+2 x)^2}\right ) \, dx \\ & = \frac {3 (625-e)}{2 (625-e-2 x)}-2 \int \frac {e^{2 x} (-626+e+2 x)}{(-625+e+2 x)^2} \, dx \\ & = \frac {3 (625-e)}{2 (625-e-2 x)}+\frac {e^{2 x}}{625-e-2 x} \\ \end{align*}
Time = 0.64 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {1875-3 e+e^{2 x} (1252-2 e-4 x)}{390625+e^2-2500 x+4 x^2+e (-1250+4 x)} \, dx=\frac {1875-3 e+2 e^{2 x}}{1250-2 e-4 x} \]
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Time = 0.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15
method | result | size |
norman | \(\frac {-{\mathrm e}^{2 x}-\frac {1875}{2}+\frac {3 \,{\mathrm e}}{2}}{{\mathrm e}-625+2 x}\) | \(23\) |
parallelrisch | \(\frac {3 \,{\mathrm e}-1875-2 \,{\mathrm e}^{2 x}}{2 \,{\mathrm e}-1250+4 x}\) | \(24\) |
risch | \(-\frac {1875}{2 \left ({\mathrm e}-625+2 x \right )}+\frac {3 \,{\mathrm e}}{2 \left ({\mathrm e}-625+2 x \right )}-\frac {{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}\) | \(41\) |
parts | \(\frac {-\frac {1875}{2}+\frac {3 \,{\mathrm e}}{2}}{{\mathrm e}-625+2 x}-\frac {{\mathrm e}^{2 x} \left ({\mathrm e}-625\right )}{{\mathrm e}-625+2 x}+\left (-{\mathrm e}+626\right ) {\mathrm e}^{-{\mathrm e}+625} \operatorname {Ei}_{1}\left (-{\mathrm e}-2 x +625\right )-{\mathrm e} \left (-\frac {{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-{\mathrm e}^{-{\mathrm e}+625} \operatorname {Ei}_{1}\left (-{\mathrm e}-2 x +625\right )\right )-\frac {626 \,{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-626 \,{\mathrm e}^{-{\mathrm e}+625} \operatorname {Ei}_{1}\left (-{\mathrm e}-2 x +625\right )\) | \(137\) |
derivativedivides | \(-\frac {1875}{2 \left ({\mathrm e}-625+2 x \right )}+\frac {3 \,{\mathrm e}}{2 \left ({\mathrm e}-625+2 x \right )}-\frac {626 \,{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-626 \,{\mathrm e}^{-{\mathrm e}+625} \operatorname {Ei}_{1}\left (-{\mathrm e}-2 x +625\right )-\frac {{\mathrm e}^{2 x} \left ({\mathrm e}-625\right )}{{\mathrm e}-625+2 x}+\left (-{\mathrm e}+626\right ) {\mathrm e}^{-{\mathrm e}+625} \operatorname {Ei}_{1}\left (-{\mathrm e}-2 x +625\right )-{\mathrm e} \left (-\frac {{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-{\mathrm e}^{-{\mathrm e}+625} \operatorname {Ei}_{1}\left (-{\mathrm e}-2 x +625\right )\right )\) | \(145\) |
default | \(-\frac {1875}{2 \left ({\mathrm e}-625+2 x \right )}+\frac {3 \,{\mathrm e}}{2 \left ({\mathrm e}-625+2 x \right )}-\frac {626 \,{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-626 \,{\mathrm e}^{-{\mathrm e}+625} \operatorname {Ei}_{1}\left (-{\mathrm e}-2 x +625\right )-\frac {{\mathrm e}^{2 x} \left ({\mathrm e}-625\right )}{{\mathrm e}-625+2 x}+\left (-{\mathrm e}+626\right ) {\mathrm e}^{-{\mathrm e}+625} \operatorname {Ei}_{1}\left (-{\mathrm e}-2 x +625\right )-{\mathrm e} \left (-\frac {{\mathrm e}^{2 x}}{{\mathrm e}-625+2 x}-{\mathrm e}^{-{\mathrm e}+625} \operatorname {Ei}_{1}\left (-{\mathrm e}-2 x +625\right )\right )\) | \(145\) |
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Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {1875-3 e+e^{2 x} (1252-2 e-4 x)}{390625+e^2-2500 x+4 x^2+e (-1250+4 x)} \, dx=\frac {3 \, e - 2 \, e^{\left (2 \, x\right )} - 1875}{2 \, {\left (2 \, x + e - 625\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.55 \[ \int \frac {1875-3 e+e^{2 x} (1252-2 e-4 x)}{390625+e^2-2500 x+4 x^2+e (-1250+4 x)} \, dx=- \frac {1875 - 3 e}{4 x - 1250 + 2 e} - \frac {e^{2 x}}{2 x - 625 + e} \]
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\[ \int \frac {1875-3 e+e^{2 x} (1252-2 e-4 x)}{390625+e^2-2500 x+4 x^2+e (-1250+4 x)} \, dx=\int { -\frac {2 \, {\left (2 \, x + e - 626\right )} e^{\left (2 \, x\right )} + 3 \, e - 1875}{4 \, x^{2} + 2 \, {\left (2 \, x - 625\right )} e - 2500 \, x + e^{2} + 390625} \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {1875-3 e+e^{2 x} (1252-2 e-4 x)}{390625+e^2-2500 x+4 x^2+e (-1250+4 x)} \, dx=\frac {3 \, e - 2 \, e^{\left (2 \, x\right )} - 1875}{2 \, {\left (2 \, x + e - 625\right )}} \]
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Time = 0.15 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {1875-3 e+e^{2 x} (1252-2 e-4 x)}{390625+e^2-2500 x+4 x^2+e (-1250+4 x)} \, dx=-\frac {3\,x+{\mathrm {e}}^{2\,x}}{2\,x+\mathrm {e}-625} \]
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